Doubling Time

Question

You deposit $500 in an account that pays 6% annual interest, compounded continuously. How long before your balance doubles?

Answer 1

We use the compound continuous formula

 

A = Pe^rt

 

where:

A = amount accumulate

P = deposit

r = interest rate

t = time

 

Since we want the balance to double, A = 2P.  The formula now becomes

 

2P = Pe^rt

 

2 = e^rt

 

We want to solve for t.  We need to use logarithms.

 

 

Natural log both sides of the equation and bring the exponent down as the coefficient of the natural log.

 

Ln(2) = rtLn(e)

 

 

Divide both sides of equation by  rLn(e).

 

t = (Ln(2)) / (rLn(e))

t = (Ln(2)) / r

 

 

Next, is to plug in these values into t.

 

r = 0.06

 

Answer 2

A = pe^rt, where A = amount, p = principal, e = Euler's constant, r = rate, t = time

 

1000 = 500e^0.06t

2 = e^0.06t

ln 2 = ln e^0.06t

ln 2 = 0.06t

0.693147 ≈ 0.06t

11.552 ≈ t